Our common understanding of the physical world deeply relies on the notion that events are ordered with respect to some time parameter, with past events serving as causes for future ones. Nonetheless, it was recently found that it is possible to formulate quantum mechanics without any reference to a global time or causal structure. The resulting framework includes new kinds of quantum resources that allow performing tasks-in particular, the violation of causal inequalities-which are impossible for events ordered according to a global causal order. However, no physical implementation of such resources is known. Here we show that a recently demonstrated resource for quantum computationthe quantum switch-is a genuine example of 'indefinite causal order'. We do this by introducing a new tool-the causal witness-which can detect the causal nonseparability of any quantum resource that is incompatible with a definite causal order. We show however that the quantum switch does not violate any causal inequality.

Quantum computers achieve a speed-up by placing quantum bits (qubits) in superpositions of different states. However, it has recently been appreciated that quantum mechanics also allows one to ‘superimpose different operations'. Furthermore, it has been shown that using a qubit to coherently control the gate order allows one to accomplish a task—determining if two gates commute or anti-commute—with fewer gate uses than any known quantum algorithm. Here we experimentally demonstrate this advantage, in a photonic context, using a second qubit to control the order in which two gates are applied to a first qubit. We create the required superposition of gate orders by using additional degrees of freedom of the photons encoding our qubits. The new resource we exploit can be interpreted as a superposition of causal orders, and could allow quantum algorithms to be implemented with an efficiency unlikely to be achieved on a fixed-gate-order quantum computer.

It is usually assumed that a quantum computation is performed by applying gates in a specific order. One can relax this assumption by allowing a control quantum system to switch the order in which the gates are applied. This provides a more general kind of quantum computing that allows transformations on blackbox quantum gates that are impossible in a circuit with fixed order. Here we show that this model of quantum computing is physically realizable, by proposing an interferometric setup that can implement such a quantum control of the order between the gates. We show that this new resource provides a reduction in computational complexity: we propose a problem that can be solved by using OðnÞ blackbox queries, whereas the best known quantum algorithm with fixed order between the gates requires Oðn 2 Þ queries. Furthermore, we conjecture that solving this problem in a classical computer takes exponential time, which may be of independent interest. A useful tool to calculate the complexity of a quantum algorithm is the blackbox model of quantum computation. In this model, the input to the computation is encoded in a unitary gate-treated as a blackbox-and the complexity of the algorithm is the number of times this gate has to be queried to solve the problem.Typically, blackbox computation is studied within the quantum circuit formalism [1]. A quantum circuit consists of a collection of wires, representing quantum systems, that connect boxes, representing unitary transformations. In this framework, wires are assumed to connect the various gates in a fixed structure; thus, the order in which the gates are applied is determined in advance and independently of the input states. It was first proposed in Ref.[2] that such a constraint can be relaxed: one can consider situations where the wires, and thus the order between gates, can be controlled by some extra variable. This is natural if one thinks of the circuit's wires as quantum systems that can be in superposition.Such "superpositions of orders" allow performing information-theoretical tasks that are impossible in the quantum circuit model: it was shown in Ref.[3] that it is possible to decide whether a pair of blackbox unitaries commute or anticommute with a single use of each unitary, whereas in a circuit with a fixed order at least one of the unitaries must be used twice. (The same task was considered in a quantum optics context in Ref. [4], where a less efficient protocol was found.)It was not known, however, whether this advantage can be translated into more efficient algorithms for quantum computing, i.e., if a quantum computer that can control the order between gates can solve a computational problem with asymptotically less resources than a quantum computer with fixed circuit structure.Here we present such a problem: given a set of n unitary matrices and the promise that they satisfy one out of n! specific properties, find which property is satisfied. The essential resource to solve this problem is the quantum control over the order of n blackboxes, first in...